Sequence

Sequences

A sequence is a list of numbers that follow a particular pattern, usually separated by commas.

Example:

$a_1, a_2, a_3, a_4, \ldots$

Each number in the sequence is called a term.

Example 1

Find the next three terms in the following sequences:

1. $5, 8, 11, 14, 17, \ldots$ Add 3 each time: Next terms: $20, 23, 26$
2. $3, 7, 6, 10, 9, \ldots$ Alternate adding 4 and subtracting 1: Next terms: $13, 12, 16$
3. $1, 2, 4, 7, \ldots$ Add 1, then 2, then 3, etc.: Next terms: $11, 16, 22$
4. $2, 9, 20, 35, \ldots$ Differences: +7, +11, +15 (increasing by 4): Next terms: $54, 77, 104$

Example 2

Given the nth term of a sequence, find the first three terms:

Let $a_n = n^2 - 1$

$a_1 = 1^2 - 1 = 0$

$a_2 = 2^2 - 1 = 3$

$a_3 = 3^2 - 1 = 8$

Example 3

Given: $k^{th}$ term = $k^2 + 4$

Find the sum of the first four terms:

$a_1 = 1^2 + 4 = 5$

$a_2 = 2^2 + 4 = 8$

$a_3 = 3^2 + 4 = 13$

$a_4 = 4^2 + 4 = 20$

Sum = $5 + 8 + 13 + 20 = 46$

Example 4

Find the nth term of the sequence:

Let the sequence be $4, 7, 10, 13, 16, \ldots$

Common difference: $d = 3$

Use the formula:

$a_n = a_1 + (n - 1)d$

Example 5

Given the sequence: $6, 4, 2, 0, -2, \ldots$

Common difference: $d = -2$

Geometric Progression (G.P.)

In a geometric sequence, each term is found by multiplying the previous term by a fixed number called the common ratio $r$.

Example:

$2, 4, 8, 16, 32, \ldots$

Here, $r = \frac{4}{2} = \frac{8}{4} = 2$

General Term of A.P.

If $a_1, a_2, a_3, \ldots, a_n$ is an arithmetic sequence with common difference $d$, then:

$a_n = a_1 + (n - 1)d$

Example 6

Sequence: $8, 9.5, 11, 12.5, 14, 15.5, \ldots$

Common difference: $d = 1.5$, First term: $a_1 = 8$

$a_n = 8 + (n - 1)(1.5) = 1.5n + 6.5$

Example 7

Given: $a_5 = 11$, $a_8 = 26$

From $a_5 = a_1 + 4d = 11$

From $a_8 = a_1 + 7d = 26$

Solve these equations to find $a_1$ and $d$

Example 8

Given:

$a_8 = a_5 + 9$

$a_{10} = 10a_2$

Use the general formula to solve and find $d$, then determine $a_{20}$

General Term of G.P.

If $G_1, G_2, G_3, \ldots, G_n$ is a geometric sequence with common ratio $r$, then:

$G_n = G_1 \cdot r^{n-1}$

Example 9

$2, 6, 18, 54, \ldots$

$r = 3$, $G_1 = 2$

$G_n = 2 \cdot 3^{n-1}$

$4, -2, 1, -0.5, 0.25, \ldots$

$r = -\frac{1}{2}$, $G_1 = 4$

$G_n = 4 \cdot \left(-\frac{1}{2}\right)^{n-1}$

Example 10

Find the 6th term of the geometric progression: 3, 6, 12, 24, ...

We are given:

First term, $a = 3$

Common ratio, $r = \frac{6}{3} = 2$

The general term of a G.P. is given by:

$a_n = ar^{n-1}$

So the 6th term is:

$a_6 = 3 \times 2^{6-1} = 3 \times 2^5 = 3 \times 32 = 96$

Answer: 96

Example 11

Find the sum of the first 5 terms of the G.P.: 2, 4, 8, 16, 32

We are given:

First term, $a = 2$

Common ratio, $r = 2$

Number of terms, $n = 5$

The sum of the first $n$ terms in a G.P. is:

$S_n = a \frac{r^n - 1}{r - 1}$, for $r \ne 1$

So,

$S_5 = 2 \times \frac{2^5 - 1}{2 - 1} = 2 \times \frac{32 - 1}{1} = 2 \times 31 = 62$

Answer: 62

Example 12

Find the 7th term of the G.P. whose first term is 5 and common ratio is 3.

We are given:

First term, $a = 5$

Common ratio, $r = 3$

Term number, $n = 7$

Using the formula:

$a_n = ar^{n-1}$

Then,

$a_7 = 5 \times 3^{7-1} = 5 \times 3^6 = 5 \times 729 = 3645$

Answer: 3645